“Harry Potter and the Prisoner of Azkaban” is the best yet of the filmed versions of novelist J.K. Rowling’s “Harry Potter” series, about a young orphan, living with hateful relatives, who discovers on his eleventh birthday that he has a literally magical heritage when he’s whisked away to Hogwarts, a school for young wizards and witches.

The previous screen incarnations of “Harry” were charming, but “Harry Potter and the Prisoner of Azkaban,” with its characters and cast both entering their teens and slightly more complex and dark story, feels at last like a movie genuinely aimed as much at adults as youngsters. It resonates in ways the earlier films did not attempt to do – and it has some of the most astounding special effects imaginable.

Azkaban is the name of the wizarding-world prison. Sirius Black (Gary Oldman), who is notorious for having a hand in the murder of Harry’s parents when Harry (Daniel Radcliffe) was a baby, has just become the first person to escape from Azkaban and it’s feared he’s going after Harry. To thwart any murder attempts and in the hopes of catching Black, the Ministry of Magic stations Dementors around the grounds of Hogwarts, over the objections of headmaster Dumbledore (Michael Gambon, taking over for the late Richard Harris who originated the role). Dementors are themselves soul-sucking monsters who inspire real terror in Harry, but once he gets to school, he finds his hands full with many other matters as well. He and his best friends Ron (Rupert Grint) and Hermione (Emma Watson) try to locate Black before he can get the jump on Harry, which leads to a number of related mysteries within Hogwarts. Meanwhile, the trio are also engaged in an effort to help their monster-loving friend Hagrid protect his hippogriff Buckbeak.

The hippogriff is a triumph of CGI – it looks like a fully dimensional living creature, despite the fact that you don’t see a lot of four-legged, horse-sized birds of prey in the real world. The Dementors are likewise impressive – while they are not a barrier to bringing children to the film, they are definitely monsters for grown-ups. Youngsters will feel courageous at watching these creatures, which move with a kind of aquatic menace through the air, as Harry finds the nerve to face them onscreen.

The performances are excellent, with Radcliffe, Watson and Grint, all back for their third go-rounds in their respective parts, bringing a new edge and intensity. Radcliffe in particular has an undercurrent of real adolescent rage that makes his decent, determined Harry more dimensional and engaging. David Thewlis is splendid as a kindly new teacher with a surprising secret, Oldman is excellent in a role that might be something out of Dickens, and the other adult cast members are thoroughly commendable.

Steve Kloves, who scripted the previous two “Harry” films as well, does a superb job of keeping everything clear, quick and urgent, condensing the massive source material without losing anything crucial. Director Alfonso Cuaron, new to the series, maintains the requisite sense of wonder while bringing in a more mature humor and sense of dread. Sound is fantastic, with all sorts of discrete effects drawing our attention to wherever something is taking place in the environment – left, right, ahead of us, behind us – and the flying scenes are much better, with a rainy Quidditch match where the players fly through the sky on broomsticks, looking greatly improved over its predecessors. Harry’s flight on Buckbeak takes us all soaring with him.

“Harry Potter and the Prisoner of Azkaban” is a wondrous workout for our imaginations – it has the thrills of good fantasy, the satisfactions of good storytelling and the joy of really effective filmmaking all bundled together in one beautiful package.